Modeling the Distribution of Ranks, Selmer Groups, and Shafarevich Tate Groups of Elliptic Curves

Using only linear algebra over Zp, we de ne a discrete probability distribution on the set of isomorphism classes of short exact sequences of Zp-modules, and then conjecture that as E varies over elliptic curves over a xed global eld k, the distribution of 0→ E(k)⊗Qp/Zp → Selp∞ E →X[p∞]→ 0 is that one. This single conjecture would explain many of the known theorems and conjectures on ranks, Selmer groups, and Shafarevich Tate groups of elliptic curves. We also prove new theorems on the arithmetic of elliptic curves that partially justify our conjecture.